Numerical Methods for Multilattices
A. Abdulle, P. Lin and A. V. Shapeev
Multiscale Modeling and Simulation, 10, 696-726 (2012).

ABSTRACT
Among the efficient numerical methods based on atomistic models, the
quasi-continuum (QC) method has attracted growing interest in recent years. The
QC method was first developed for crystalline materials with Bravais lattice and
was later extended to multilattices [Tadmor et al., Phys. Rev. B, 59 (1999), pp.
235-245]. Another existing numerical approach to modeling multilattices is
homogenization. In the present paper we review the existing numerical methods
for multilattices and propose another concurrent macro-to-micro method in the
numerical homogenization framework. We give a unified mathematical formulation
of the new and the existing methods and show their equivalence. We then consider
extensions of the proposed method to time-dependent problems and to random
materials.